Time-dependent Coordinate System Research Articles

Characterization of events in the Rössler system using singular value decomposition

Chaotic systems change their behavior through topological events such as creation/annihilation and bifurcation. These can be characterized by defining a tangent phase space which measures the first-order response of stable limit cycles to a change in an external variable. If the period of the limit cycle is constant, then the tangent phase space response can be formulated as a boundary-value problem, which is dependent upon a previously calculated limit cycle. If the period is not constant, the tangent phase space will contain an unknown linear drift in time. This can be analytically removed by transforming into a time-dependent coordinate system in which one variable points in the direction of the instantaneous velocity in phase space. The remaining variables can then be decoupled from this motion, and will satisfy a linear system of differential equations subject to periodic boundary conditions. The solutions of these equations at bifurcation events can be analyzed using singular value decomposition of two matrices, one of which contains interactions within the limit cycle, while the other contains interactions with the changing external variable. Collectively, these two decompositions allow us to uniquely characterize any topological event. The method is applied to period-doubling and turning-points of limit cycles in the Rössler system, where it confirms previous work done on the Zeeman Catastrophe Machine. It is also applied to bifurcations of equilibria in the Rössler system, where it allows us to distinguish between Andronov-Hopf and fold-Hopf bifurcations.

3D Numerical Simulation of the Interaction between Waves and a T-Head Groin Structure

The aim of coastal structures for the defense from erosion is to modify the hydrodynamic fields that would naturally occur with the wave motion, to produce zones of sedimentation of solid material, and to combat the recession of the coastline. T-head groin-shaped structures are among the most adopted in coastal engineering. The assessment of the effectiveness of such structures requires hydrodynamic study of the interaction between wave motion and the structure. Hydrodynamic phenomena induced by the interaction between wave motion and T-head groin structures have three-dimensionality features. The aim of the paper is to propose a new three-dimensional numerical model for the simulation of the hydrodynamic fields induced by the interaction between wave fields and coastal structures. The proposed model is designed to represent complex morphologies as well as coastal structures inside the domain. The numerical scheme solves the three-dimensional Navier–Stokes equations in a contravariant formulation, on a time-dependent coordinate system, in which the vertical coordinate varies over time to follow the free-surface elevation. The main innovative element of the paper consists in the proposal of a new numerical scheme that makes it possible to simulate flows around structures with sharp-cornered geometries. The proposed numerical model is validated against a well-known experimental test-case consisting in a wave train approaching a beach (non-parallel with the wave front), with the presence of a T-head groin structure. A detailed comparison between numerical and experimental results is shown.

Numerical and Experimental Investigation of Wave Overtopping of Barriers

We present a study of wave overtopping of barriers. The phenomenon of the wave overtopping over emerged structures is reproduced both numerically and experimentally. The numerical simulations are carried out by a numerical scheme for three-dimensional free-surface flows, which is based on the solution of the Navier–Stokes equations in a novel integral form on a time-dependent coordinate system. In the adopted numerical scheme, a novel wet–dry technique, based on the exact solution of the Riemann problem over the dry bed, is proposed. The experimental tests are carried out by adopting a nonintrusive and continuous-in-space image-analysis technique, which is able to properly identify the free surface even in very shallow waters or breaking waves. A comparison between numerical and experimental results, for several wave and water-depth conditions, is shown.

Nonlinear waves and nearshore currents over variable bathymetry in curve-shaped coastal areas

Irregular coastlines and variable bathymetry produce nonlinear effects on wave propagation which play a significant role on the formation of nearshore currents. To protect the coastline from the erosional action of nearshore currents, it is usual to adopt coastal defence works such as submerged breakwaters. If properly designed, they give rise to circulation patterns capable to induce sedimentation of suspended material at the nearshore region. To numerically simulate the hydrodynamic effects of submerged breakwaters in irregular coastal areas, we use a numerical model which is based on an integral contravariant formulation of the three-dimensional Navier–Stokes equations in a time-dependent coordinate system. These equations are numerically solved by a non-hydrostatic shock-capturing numerical scheme which is able to simulate the wave propagation from deep water to the shoreline, including the surf zone and swash zone.

A new three-dimensional finite-volume non-hydrostatic shock-capturing model for free surface flow

In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena (wave refraction, diffraction, shoaling and breaking) is proposed. The model is based on an integral formulation of the Navier-Stokes equations which are solved on a time dependent coordinate system: a coordinate transformation maps the varying coordinates in the physical domain to a uniform transformed space. The equations of motion are discretized by means of a finite-volume shock-capturing numerical procedure based on high order WENO reconstructions. The solution procedure for the equations of motion uses a third order accurate Runge-Kutta (SSPRK) fractional-step method and applies a pressure corrector formulation in order to obtain a divergence-free velocity field at each stage. The proposed model is validated against several benchmark test cases.

Euler and Navier–Stokes equations in a new time-dependent helically symmetric system: derivation of the fundamental system and new conservation laws

The present contribution is a significant extension of the work by Kelbin et al. (J. Fluid Mech., vol. 721, 2013, pp. 340–366) as a new time-dependent helical coordinate system has been introduced. For this, Lie symmetry methods have been employed such that the spatial dependence of the originally three independent variables is reduced by one and the remaining variables are: the cylindrical radius $r$ and the time-dependent helical variable $\unicode[STIX]{x1D709}=(z/\unicode[STIX]{x1D6FC}(t))+b\unicode[STIX]{x1D711}$, $b=\text{const.}$ and time $t$. The variables $z$ and $\unicode[STIX]{x1D711}$ are the usual cylindrical coordinates and $\unicode[STIX]{x1D6FC}(t)$ is an arbitrary function of time $t$. Assuming $\unicode[STIX]{x1D6FC}=\text{const.}$, we retain the classical helically symmetric case. Using this, and imposing helical invariance onto the equation of motion, leads to a helically symmetric system of Euler and Navier–Stokes equations with a time-dependent pitch $\unicode[STIX]{x1D6FC}(t)$, which may be varied arbitrarily and which is explicitly contained in all of the latter equations. This has been conducted both for primitive variables as well as for the vorticity formulation. Hence a significantly extended set of helically invariant flows may be considered, which may be altered by an external time-dependent strain along the axis of the helix. Finally, we sought new conservation laws which can be found from the helically invariant Euler and Navier–Stokes equations derived herein. Most of these new conservation laws are considerable extensions of existing conservation laws for helical flows at a constant pitch. Interestingly enough, certain classical conservation laws do not admit extensions in the new time-dependent coordinate system.

Numerical Analysis of the Internal Flow in an Annular Flow Channel Type Oil Damper

The purpose of the present study is to clarify the fluid flow of an oil damper through numerical analysis in order to obtain an exact value of the damping coefficient of an oil damper. The finite difference method (FDM) was used to solve the governing equation of the fluid flow generated by a moving piston. Time steps evolved according to the fractional step method, and the arbitrary Lagrangian–Eulerian (ALE) method was adopted for the moving boundary. In order to stabilize the computation in the moving boundary problem, a masking method with a single block grid system was used. In other words, algebraic grid generation using a stretching function was used for the moving piston in the cylinder of the oil damper. The time-dependent coordinate system in the physical domain, which coincides with the contour of the moving boundary, is transformed into a fixed rectangular coordinate system in the computational domain. The computational results were compared with experimentally obtained results and the approximate analytical solution. The results of the present analysis exhibit good agreement with the experimental results over various widths of the annular flow channel between the piston and cylinder.

Numerical Analysis of the Internal Flow in an Oil Damper (Analysis Based on the Two-Dimensional Cylindrical Coordinates to the Damper with Annular Cross-Sectional Flow Channel)

The purpose of this study is to know the fluid flow of an oil damper by a numerical analysis, so as to obtain the exact values of the damping coefficient of an oil damper. The Finite Difference Method (FDM) was used for solving the governing equation of the fluid caused by the moving piston. Each time step proceeded in the Fractional Step method, and the Arbitrary Lagrangian-Eulerian (ALE) method was adopted for the moving boundary. In order to stabilize the computation in the moving boundary problem, we employed the masking method with a single block grid system. That is, the algebraic grid generation using stretching function was used for the moving piston. The time dependent coordinate system in physical domain which coincides with a contour of moving boundary is transformed into a fixed rectangular coordinate system in computational domain. The computational results were compared with the experimental ones and the approximate analytical solution. The results of present analysis show good agreement with the experimental results over a wide range of piston diameter.

Timing and geodesy

Geodetic science (geodesy) is that branch of science concerned with determining the size, shape, and gravitational field of the earth, and precisely mapping its surface features. Astronomic observations provide essential information in the geodetic process. Both the earth-based observing stations and the observed extraterrestrial bodies (stars, planets, natural and artificial earth satellites) are in motion, creating time-dependent coordinate systems and making epoch, interval, and synchronization important. Modern geodesy uses the most sophisticated optical and electronic equipment in measurements, and the timing requirements have become increasingly stringent. Systems that require accurate timing often have potential in the determination of time. Lunar laser ranging, a currently developing technique with many exciting geodetic implications, may develop into a practical method for the determination of Universal Time (UT1) to a new order of accuracy.